Imaginary unit

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x² + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i.

Imaginary numbers are an important mathematical concept; they extend the real number system ${\displaystyle \mathbb {R} }$ to the complex number system ${\displaystyle \mathbb {C} ,}$ in which at least one

.

There are two complex square roots of −1: i and −i, just as there are two complex

double square root

).

In contexts in which use of the letter i is ambiguous or problematic, the letter j is sometimes used instead. For example, in

Terminology

Square roots of negative numbers are called imaginary because in

A unit is an undivided whole, and unity or the unit number is the number one (1).

Definition

The powers of i are cyclic:
${\displaystyle \ \vdots }$
${\displaystyle \ i^{-4}={\phantom {-}}1{\phantom {i}}}$
${\displaystyle \ i^{-3}={\phantom {-}}i{\phantom {1}}}$
${\displaystyle \ i^{-2}=-1{\phantom {i}}}$
${\displaystyle \ i^{-1}=-i{\phantom {1}}}$
${\displaystyle \ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}}$
${\displaystyle \ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}}$
${\displaystyle \ \ i^{2}\ =-1{\phantom {i}}}$
${\displaystyle \ \ i^{3}\ =-i{\phantom {1}}}$
${\displaystyle \ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}}$
${\displaystyle \ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}}$
${\displaystyle \ \ i^{6}\ =-1{\phantom {i}}}$
${\displaystyle \ \ i^{7}\ =-i{\phantom {1}}}$
${\displaystyle \ \vdots }$

The imaginary unit i is defined solely by the property that its square is −1:

With i defined this way, it follows directly from algebra that i and −i are both square roots of −1.

Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by treating i as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of i² with −1). Higher integral powers of i are thus

and so on, cycling through the four values 1, i, −1, and −i. As with any non-zero real number, i⁰ = 1.

As a complex number, i can be represented in

(or angle) of ${\displaystyle {\tfrac {\pi }{2}}}$

real axis

).

i vs. −i

Being a

of each other. Although the two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled +i (or simply i) and the other is labelled −i, though which is which is inherently ambiguous.

The only differences between +i and −i arise from this labelling. For example, by convention +i is said to have an argument of ${\displaystyle +{\tfrac {\pi }{2}}}$ and −i is said to have an argument of ${\displaystyle -{\tfrac {\pi }{2}},}$ related to the convention of labelling orientations in the

A more formal expression of this indistinguishability of +i and −i is that, although the complex

of the complex numbers ${\displaystyle \mathbb {C} }$ that keep each real number fixed, namely the identity and .

Matrices

Using the concepts of matrices and matrix multiplication, complex numbers can be represented in linear algebra. The real unit 1 and imaginary unit i can be represented by any pair of matrices I and J satisfying I² = I, IJ = JI = J, and J² = −I. Then a complex number a + bi can be represented by the matrix aI + bJ, and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic.

The most common choice is to represent 1 and i by the 2 × 2 identity matrix I and the matrix J,

Then an arbitrary complex number a + bi can be represented by:

More generally, any real-valued 2 × 2 matrix with a

for spatial dimensions.

Root of x² + 1

Polynomials (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose coefficients are real numbers form a ring, denoted ${\displaystyle \mathbb {R} [x],}$ an algebraic structure with addition and multiplication and sharing many properties with the ring of integers.

The polynomial ${\displaystyle x^{2}+1}$ has no real-number

, but the set of all real-coefficient polynomials divisible by ${\displaystyle x^{2}+1}$ forms an ideal, and so there is a quotient ring ${\displaystyle \mathbb {R} [x]/\langle x^{2}+1\rangle .}$ This quotient ring is isomorphic to the complex numbers, and the variable ${\displaystyle x}$ expresses the imaginary unit.

Graphic representation

The complex numbers can be represented graphically by drawing the real number line as the horizontal axis and the imaginary numbers as the vertical axis of a Cartesian plane called the complex plane. In this representation, the numbers 1 and i are at the same distance from 0, with a right angle between them. Addition by a complex number corresponds to translation in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every similarity transformation of the plane can be represented by a complex-linear function ${\displaystyle z\mapsto az+b.}$

Geometric algebra

In the geometric algebra of the Euclidean plane, the geometric product or quotient of two arbitrary vectors is a sum of a scalar (real number) part and a bivector part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector is a negative scalar.

The quotient of a vector with itself is the scalar 1 = u/u, and when multiplied by any vector leaves it unchanged (the

More generally, in the geometric algebra of any higher-dimensional Euclidean space, a unit bivector of any arbitrary planar orientation squares to −1, so can be taken to represent the imaginary unit i.

Proper use

The imaginary unit was historically written ${\textstyle {\sqrt {-1}},}$ and still is in some modern works. However, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation ${\textstyle {\sqrt {x}}}$ is reserved either for the principal square root function, which is defined for only real x ≥ 0, or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results:^[7]

Generally, the calculation rules ${\textstyle {\sqrt {x{\vphantom {ty}}}}\cdot \!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x\cdot y{\vphantom {ty}}}}}$ and ${\textstyle {\sqrt {x{\vphantom {ty}}}}{\big /}\!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x/y}}}$ are guaranteed to be valid for real, positive values of x and y only.^[8][9][10]

When x or y is real but negative, these problems can be avoided by writing and manipulating expressions like ${\textstyle i{\sqrt {7}}}$, rather than ${\textstyle {\sqrt {-7}}}$. For a more thorough discussion, see square root and branch point.

Properties

As a complex number, the imaginary unit follows all of the rules of complex arithmetic.

Imaginary integers and imaginary numbers

When the imaginary unit is repeatedly added or subtracted, the result is some integer times the imaginary unit, an imaginary integer; any such numbers can be added and the result is also an imaginary integer:

Thus, the imaginary unit is the generator of a group under addition, specifically an infinite cyclic group.

The imaginary unit can also be multiplied by any arbitrary real number to form an imaginary number. These numbers can be pictured on a number line, the imaginary axis, which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally.

Gaussian integers

Integer sums of the real unit 1 and the imaginary unit i form a

. The sum, difference, or product of Gaussian integers is also a Gaussian integer:

Quarter-turn rotation

When multiplied by the imaginary unit i, any arbitrary complex number in the complex plane is rotated by a quarter turn (${\displaystyle {\tfrac {1}{2}}\pi }$ radians or 90°)

. When multiplied by −i, any arbitrary complex number is rotated by a quarter turn clockwise. In polar form:

In rectangular form,

Integer powers

The powers of i repeat in a cycle expressible with the following pattern, where n is any integer:

Thus, under multiplication, i is a generator of a cyclic group of order 4, a discrete subgroup of the continuous circle group of the unit complex numbers under multiplication.

Written as a special case of Euler's formula for an integer n,

With a careful choice of

]

Roots

Just like all nonzero complex numbers, ${\textstyle i=e^{\pi i/2}}$ has two distinct square roots which are additive inverses. In polar form, they are

In rectangular form, they are^[a]

Squaring either expression yields

The three cube roots of i are^[12]

For a general positive integer n, the n-th roots of i are, for k = 0, 1, ..., n − 1,

The value associated with k = 0 is the principal n-th root of i. The set of roots equals the corresponding set of roots of unity rotated by the principal n-th root of i. These are the vertices of a regular polygon inscribed within the complex unit circle.

Exponential and logarithm

The

function relates complex addition in the domain to complex multiplication in the codomain. Real values in the domain represent scaling in the codomain (multiplication by a real scalar) with 1 representing multiplication by e, while imaginary values in the domain represent rotation in the codomain (multiplication by a unit complex number) with i representing a rotation by 1 radian. The complex exponential is thus a periodic function in the imaginary direction, with period 2πi and image 1 at points 2kπi for all integers k, a real multiple of the lattice of imaginary integers.

The complex exponential can be broken into even and odd components, the hyperbolic functions cosh and sinh or the trigonometric functions cos and sin:

Euler's formula decomposes the exponential of an imaginary number representing a rotation:

The quotient coth z = cosh z / sinh z, with appropriate scaling, can be represented as an infinite

$${\displaystyle \pi \coth \pi z=\lim _{n\to \infty }\sum _{k=-n}^{n}{\frac {1}{z+ki}}.}$$

Other functions based on the complex exponential are well-defined with imaginary inputs. For example, a number raised to the ni power is:

$${\displaystyle x^{ni}=\cos(n\ln x)+i\sin(n\ln x).}$$

Because the exponential is periodic, its inverse the

Functions depending on the complex logarithm therefore depend on careful choice of branch to define and evaluate clearly.

For example, if one chooses any branch where ${\displaystyle \ln i={\tfrac {1}{2}}\pi i}$ then when x is a positive real number,

$${\displaystyle \log _{i}x=-{\frac {2i\ln x}{\pi }}.}$$

Factorial

The factorial of the imaginary unit i is most often given in terms of the gamma function evaluated at 1 + i:^[15]

The magnitude and argument of this number are:^[16]

See also

• Hyperbolic unit
• Right versor
  in quaternions

Notes

References

Further reading

• Nahin, Paul J. (1998). An Imaginary Tale: The story of i [the square root of minus one]. Chichester: Princeton University Press.
  ISBN 0-691-02795-1
  – via Archive.org.

External links

• Euler, Leonhard. "Imaginary Roots of Polynomials". at "Convergence". mathdl.maa.org. Mathematical Association of America. Archived from the original on 13 July 2007.